Simple way to add button to UI
You basically need to combine two of the Templates in Text Editor that ship with Blender:

An Operator and a panel. In the panel's draw() method, you add an operator to the layout, which will be represented by a clickable button. Clicking it will run the operator's execute() method.

Here is an example how interface and actions play together:

http://wiki.blender.org/index.php/Dev:2.5/Py/Scripts/Cookbook/Panels_and_Operators/Ping_Pong